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A Probabilistic Approach to the Interpolation Error | ||
| Computational Sciences and Engineering | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 23 فروردین 1405 | ||
| نوع مقاله: Original Article | ||
| شناسه دیجیتال (DOI): 10.22124/cse.2026.33090.1163 | ||
| نویسندگان | ||
| Sadia Kumail Awad1؛ Ali Tavakoli* 2 | ||
| 1Polytechnic College/Al-Qadisiyah, Al-Furat Al-Awsat Technical University, Iraq | ||
| 2University of Mazandaran | ||
| چکیده | ||
| Polynomial interpolation is a fundamental tool in numerical analysis, with its accuracy classically characterized by the Lagrange remainder term. This term involves an unknown mean value point ξ∈[a,b], which depends on x and the function f. Consequently, while the formula provides an exact error representation, its practical utility for a priori error estimation is limited, as determining a sharp, computable bound for high-order derivatives is often challenging. This paper introduces a novel probabilistic framework to address this longstanding limitation. Instead of treating ξ as an unknown deterministic value, we model its location probabilistically. By considering the interpolation nodes as random variables or analyzing the distribution of ξ for a fixed x, we derive a statistical estimate for its expected value. This approach allows us to propose a specific, computable value for ξ that provides a highly accurate estimate of the actual interpolation error. The theoretical findings are substantiated with several numerical examples. These experiments demonstrate that our probabilistic estimate of the remainder term consistently aligns with the true error, offering a practical and powerful alternative to traditional worst-case error bounds. This method provides a new perspective on error analysis in approximation theory, bridging deterministic numerical methods with probabilistic techniques. | ||
| کلیدواژهها | ||
| Polynomial interpolation؛ Chebyshev nodes؛ Trapezoidal rule | ||
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آمار تعداد مشاهده مقاله: 8 |
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